Subgroups — C₂×C₆₇×Q₁₂₈ #30692991

Order 17152.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×Q₁₂₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₆₇	—	2	2	no	—
5	Q₆₄×C₆₇	—	2	2	no	—
6	Q₆₄×C₆₇	C₂²	1	4	yes	—
7	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
8	C₆₄×C₆₇	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₆₇	—	4	2	no	—
10	Q₃₂×C₆₇	—	2	4	no	—
11	Q₃₂×C₆₇	—	4	2	no	—
12	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
13	C₃₂×C₆₇	D₄	1	1	yes	characteristic
14	C₃₂×C₆₇	C₂³	1	1	yes	characteristic
15	C₂×Q₈×C₆₇	—	8	2	no	—
16	Q₁₆×C₆₇	—	4	4	no	—
17	Q₁₆×C₆₇	—	8	2	no	—
18	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
19	C₁₆×C₆₇	D₈	1	1	yes	characteristic
20	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
21	Q₈×C₆₇	—	8	4	no	—
22	Q₈×C₆₇	—	16	2	no	—
23	C₂×C₄×C₆₇	—	16	2	no	—
24	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
25	C₈×C₆₇	D₁₆	1	1	yes	characteristic
26	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
27	C₂²×C₆₇	D₃₂	1	1	yes	characteristic
28	C₄×C₆₇	—	16	4	no	—
29	C₄×C₆₇	D₃₂	1	1	yes	characteristic
30	C₄×C₆₇	C₂×D₁₆	1	1	yes	characteristic
31	C₂×Q₁₂₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₆₇	Q₁₂₈	1	2	yes	direct product factor
33	C₂×C₆₇	C₂×D₃₂	1	1	yes	characteristic
34	C₂×Q₆₄	C₂×C₆₇	1	2	yes	—
35	Q₁₂₈	C₂×C₆₇	1	4	yes	direct product factor
36	C₂×C₆₄	C₂×C₆₇	1	1	yes	characteristic
37	C₆₇	C₂×Q₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×Q₃₂	—	2	2	no	—
39	Q₆₄	—	2	2	no	—
40	Q₆₄	C₂²×C₆₇	1	4	yes	—
41	C₂×C₃₂	C₂²×C₆₇	1	1	yes	characteristic
42	C₆₄	C₂²×C₆₇	1	2	yes	—
43	C₂×Q₁₆	—	4	2	no	—
44	Q₃₂	—	2	4	no	—
45	Q₃₂	—	4	2	no	—
46	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
47	C₃₂	D₄×C₆₇	1	1	yes	characteristic
48	C₃₂	C₂³×C₆₇	1	1	yes	characteristic
49	C₂×Q₈	—	8	2	no	—
50	Q₁₆	—	4	4	no	—
51	Q₁₆	—	8	2	no	—
52	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
53	C₁₆	D₈×C₆₇	1	1	yes	characteristic
54	C₁₆	C₂×D₄×C₆₇	1	1	yes	characteristic
55	Q₈	—	8	4	no	—
56	Q₈	—	16	2	no	—
57	C₂×C₄	—	16	2	no	—
58	C₂×C₄	D₁₆×C₆₇	1	1	yes	characteristic
59	C₈	D₁₆×C₆₇	1	1	yes	characteristic
60	C₈	C₂×D₈×C₆₇	1	1	yes	characteristic
61	C₂²	D₃₂×C₆₇	1	1	yes	characteristic
62	C₄	—	16	4	no	—
63	C₄	D₃₂×C₆₇	1	1	yes	characteristic
64	C₄	C₂×D₁₆×C₆₇	1	1	yes	characteristic
65	C₂	C₆₇×Q₁₂₈	1	2	yes	direct product factor
66	C₂	C₂×D₃₂×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₆₇
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₂×C₈×C₆₇ ◁ C₂×C₁₆×C₆₇ ◁ C₂×C₃₂×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×Q₁₂₈	C₂	direct (N × H)
31. C₂×Q₁₂₈	C₆₇	direct (N × H)
32. C₂×C₆₇	Q₁₂₈	direct (N × H)
35. Q₁₂₈	C₂×C₆₇	direct (N × H)
37. C₆₇	C₂×Q₁₂₈	direct (N × H)
65. C₂	C₆₇×Q₁₂₈	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₆₄×C₆₇	2
C₆₇×Q₁₂₈	2
C₂×C₆₄×C₆₇	2
C₂×Q₁₂₈	67

Subgroups — C2×C67×Q128 #30692991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×Q₁₂₈ #30692991